Definition of metastability

Question

I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is metastable".

What does "metastable" mean? The Wikipedia article on that term explains it only in the context of physical systems:

Metastability describes the behaviour of certain physical systems that can exist in long lived states that are less stable than the system's most stable state.

I understand that a sequence $x_n$ can be viewed as a (discrete-time) dynamical system, but I need a precise mathematical definition of this property that sequences can have.

Answer 1

See this paper by Avigad/Dean/Rute or this other paper by them. I'm not sure (haven't looked very hard into the matter), but the term metastable in this context may be something originally coined by Tao, perhaps beginning with this paper by Tao, although your latest comment suggests the term originated with someone else.

I glanced at Tao's blog on soft and hard analysis, but I'd have to spend more time than I have at the moment (or even the next few days, due to upcoming deadlines at work) to get a good sense of what metastable convergence means. [Side note: Off the top of my head, I know a couple of other examples of soft/hard analysis examples that are not mentioned in Tao's blog. One is E. H. Moore's notion of relatively uniform convergence for a series of functions, and another is the classification of nowhere dense sets by generalized notions of porosity (such as I observe in the middle of p. 574 of this paper).] A possibly good place to look for pointers to expository items (conference slides, class notes, etc.) about what metastability of sequences means is Jeremy Avigad's homepage, but for some reason I can't get his webpage to load right now.

Maybe by posting this (as you suggested in a comment) someone who knows more than I do will chime in, as my answer will bump your question back to the top of the questions for others to see.

Answer 2

Metastability has many interpretations, depending upon which field you are working in. In dynamical systems/Markov chain literature, metastability has been associated with states (or regions in phase space) where a typical trajectory will spend a lot of time, but that state (or region of phase space) is not an attractor (or a sink or so on).

To give you a practical example: Properties of protein molecules can be understood in terms of conformations of such molecules: i.e. millions of different ways in which the atoms can arrange themselves. Very few of such conformations are actually seen in reality, and those states are called metastable states.

Here's a paper that talks more about it:http://biocomputing.mi.fu-berlin.de/publications/ScHu03.pdf

Precise mathematical definition is hard to come by, but imagine partitioning your phase-space into n different subsets. Such a partition can be done in infinite many ways, but a metastable partition would follow certain rules. For example if the phase space subsets are $B_1$,$B_2$,...$B_n$, where $X=B_1 \cup B_2 ... \cup B_n$, $X$ being the state space, each $B_i$ metastable and $F$ the system propagator in time, then you require that at time step k, the probabilities satisfy:

$P(x_{k+1}\in B_j | x_k\in B_i) \ll 1, $ if $i \neq j$ And

$P(x_{k+1}\in B_j | x_k\in B_i) \approx 1, $ if $i = j$

where $x_{x+1}=F(x_k)$